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%I #88 Feb 21 2023 17:41:42
%S 0,1,5,13,29,51,75,99,135,175,207,251,315,409,481,537,613,685,717,765,
%T 845,957,1097,1237,1377,1545,1665,1797,1965,2203,2371,2491,2647,2783,
%U 2815,2863,2943,3055,3195,3339,3503,3727,3943,4199,4471,4839,5163,5479,5759,6055,6215,6365,6597,6917,7321,7753,8161
%N D-toothpick sequence of the third kind starting with a single toothpick.
%C This is a cellular automaton of forking paths to 135 degrees which uses elements of three sizes: toothpicks of length 1, D-toothpicks of length 2^(1/2) and D-toothpicks of length 2^(1/2)/2. Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.
%C On the infinite square grid we start with no elements.
%C At stage 1, place a single toothpick on the paper, aligned with the y-axis. The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited.
%C The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A220501) give the number of toothpicks or D-toothpicks added at n-th stage.
%C It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or fractal-like) behavior related to powers of 2. Note that for some values of n we can see an internal growth.
%C The structure contains eight wedges. Each vertical wedge (see A220520) also contains infinitely many copies of the oblique wedges. Each oblique wedge (see A220522) also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.
%C The structure is mysterious: it contains at least 59 distinct internal regions (or polygonal pieces), for example: one of the concave octagons appears for first time at stage 223. The largest known polygon is a concave 24-gon. The exact number of distinct polygons is unknown.
%C Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9-gon and a pentagon. These subsets have a surprising connection with the Sierpinski triangle A047999, but the pattern is more complex.
%C Apparently this cellular automaton has the most complex structure of all the toothpick structures that have been studied (see illustrationsm also the illustrations of the wedges in the entries A220520 and A220522).
%C The structure contains at least 69 distinct polygonal pieces. The largest known polygon is a concave 24-gon of area 95/2 = 47.5 which appears for first time at stage 879. - _Omar E. Pol_, Feb 10 2018
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H Ayliean, <a href="https://www.youtube.com/watch?v=8VzcModlh4w">2 Minute Mathematical Meditation</a>, Youtube video (2021).
%H Leonid Broukhis, <a href="http://ioccc.org/1995/leo.c">A program</a> generating variations of the "D-toothpick" pattern, Oct. 1995; <a href="http://ioccc.org/1995/leo.hint">(remarks)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp060.jpg">Illustration of initial terms</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp070.jpg">Illustration of the structure after 17 stages</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%Y Cf. A139250, A172310, A194270, A194276, A194432, A194434, A194440, A194442, A194444, A194700, A220520, A220522.
%K nonn
%O 0,3
%A _Omar E. Pol_, Dec 15 2012
%E Terms a(23) and beyond from _David Applegate_'s movie version. - _Omar E. Pol_, Feb 10 2018
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